where \(|00\rangle _{L_2}\)  represents that \(s_1\)  is in the spin up state and Charlie's pointer variable associated with the up state, and \(|11\rangle _{L_1}\)  corresponds to the spin down state for \(s_1\)  and Charlie's pointer variable. Similar meanings for \(|00\rangle _{L_2}\)  and \(|11\rangle _{L_2}\)  for \(s_2\)  and Debbie's pointer variable. The key point of  [1]} is to assume there exists a joint probability \(p(a_1a_2b_1b_2)\) , where \(a_1, a_2\in \lbrace 0, 1\rbrace \)  are the measurement results corresponding to Alice's choice of two types of measurement operations and \(b_1, b_2\in \lbrace 0, 1\rbrace \)  are the measurement results of Bob. Alice can choose to either measure \(L_1\)  with projection operator \(A_1\) , or with projection operator \(A_2\) . Here \(A_1\)  is representedIn [1]} the two types of operations for Alice are defined as \(\mathcal {A}_1=|00\rangle \langle 00|_{L_1}-|11\rangle \langle 11|_{L_1}, i\in \lbrace 0,1\rbrace \) , and \(\mathcal {A}_2=|00\rangle \langle 11|_{L_1}-|00\rangle \langle 11|_{L_1}\) . Here we use the spectral decomposition theorem to decompose \(\mathcal {A}_1\)  into projection operators, and represent it by \(A_1\) . Similar approach for the definition of \(A_2\) . There is an important difference here compared to the setup in [3]}. Here \(a_1, a_2\)  are results for Alice from completed measurement \(A_1, A_2\)  respectively, whereas in [3]} \(a, c\)  are measurement results for Alice and Charlie respectively. The issue of \(c\)  as result of Charlie's “pre-measurement" in [3]} does not exist here in [1]}. by \(|\phi _i\rangle \langle \phi _i|_{L_1}\) , and \(|\phi _0\rangle =|00\rangle \)  for \(a_1=0\)  or \(|\phi _1\rangle =|11\rangle \)  for \(a_1=1\) . \(A_2\)  is chosen to be \(A_2=|\chi _0\rangle \langle \chi _0|\)  for \(a_2=0\)  or \(A_2=|\chi _1\rangle \langle \chi _1|\)  for \(a_2=1\) , where
\(|\chi _i\rangle = \frac{1}{\sqrt{2}}((-1)^i|00\rangle _{L_1} + |11\rangle _{L_1}).\) 
